Would you differentiate it and then show that the derivative must be greater than 0 for all values of x. Then find a point, which gives a negative value of and a point that gives a positive value of .

Would the answer always be 3. Could there be a case, where it is not 3.

For 1, I would do as you said - the derivative is . Now all you need to do is find one point where f is negative and one where it is positive.

For 2, an example of a 3rd degree polynomial which doesn't have 3 real roots is , which has one real root = -1 and two complex roots -

In this example, differentiate the polynomial and find how many zeroes the derivative has (this is easier - it's just a quadratic in x..), and then how many extremal points. This should tell you how many zeroes the polynomial has (why?)